1

Spatially and temporally non-uniform plasmas: microdischarges from the perspective of

molecules in a packed bed plasma reactor

K. van ‘t Veer1,2,*, S. van Alphen1,3, A. Remy2,4, Y. Gorbanev1, N. De Geyter4, R. Snyders3,5, F.

Reniers2, A. Bogaerts1,*

1University of Antwerp, Department of Chemistry, Research Group PLASMANT, Universiteitsplein 1,

2610 Wilrijk-Antwerp, Belgium

2Université Libre de Bruxelles, Faculty of Sciences, Chemistry of Surfaces, Interfaces and

Nanomaterials, CP255, Avenue F. D. Roosevelt 50, B-1050 Brussels, Belgium

3Université de Mons, Plasma Surface Interaction Chemistry, Place du Parc 23, 7000 Mons, Belgium

4Ghent University, Faculty of Engineering and Architecture, Department of Applied Physics,

Research Unit Plasma Technology (RUPT), Sint-Pietersnieuwstraat 41 B4, 9000 Ghent, Belgium

5Materia Nova Research Center, 1 Avenue Nicolas Copernic, B 7000 Mons, Belgium

*E-mail: kevin.vantveer@uantwerpen.be, annemie.bogaerts@uantwerpen.be

Abstract

Dielectric barrier discharges (DBDs) typically operate in the filamentary regime and thus exhibit great

spatial and temporal non-uniformity. In order to optimize DBDs for various applications, such as in

plasma catalysis, more fundamental insight is needed. Here, we consider how the millions of

microdischarges, characteristic for a DBD, influence individual gas molecules. We use a Monte Carlo

approach to determine the number of microdischarges to which a single molecule would be exposed,

by means of particle tracing simulations through a full-scale packed bed DBD reactor, as well as an

empty DBD reactor. We find that the fraction of microdischarges to which the molecules are exposed

can be approximated as the microdischarge volume over the entire reactor gas volume. The use of this

concept provides good agreement between a plasma-catalytic kinetics model and experiments for

plasma-catalytic NH3 synthesis. We also show that the concept of the fraction of microdischarges

indicates the efficiency by which the plasma power is transferred to the gas molecules. This generalised

concept is also applicable for other spatially and temporally non-uniform plasmas.

1. Introduction

Plasma catalysis is gaining a lot of interest for many applications, but the underlying mechanisms are

far from being understood [1]. The most common plasma source in plasma catalysis is a dielectric

barrier discharge (DBD) [2], which typically operates in the filamentary regime. This means that the

plasma exhibits a great spatial and temporal non-uniformity, in the form of millions of microdischarges

taking place throughout the whole reactor [3]. A better understanding of how these millions of

microdischarges interact with the individual molecules in the plasma is very important, considering the

complex reaction kinetics associated with typical gas conversion processes [4]–[6]. These complex

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chemistries are typically modelled by so-called 0D or global plasma models, in which no explicit spatial

dependency of the plasma is taken into account [7].

In order to capture the characteristics of a filamentary DBD in 0D plasma kinetics models, various

authors have adopted a certain microdischarge frequency [8]–[12], and later specifically applied a so-

called volume-corrected microdischarge frequency [13], [14]. The latter takes into account that a

molecule does not experience all the microdischarges due to the stochastic behaviour of these

microdischarges [15]. This quantity was defined as the microdischarge volume over the reactor volume

[13], [14].

In this paper we substantiate this idea, as the fraction of microdischarges to which individual molecules

are exposed, and we employ a Monte Carlo approach to calculate this fraction of microdischarges. The

Monte Carlo calculations consider particle trajectories through a reactor volume, based on a full-scale

flow model of a packed bed and empty DBD reactor, which will be discussed in detail. Furthermore,

we report ICCD images to substantiate the assumptions underlying our calculations and we use a plasma

kinetic model and plasma catalysis experiments to validate our concept of the fraction of

microdischarges. Finally, we discuss the general applicability of our findings. The fraction of

microdischarges can be related to the efficiency by which the plasma power is transferred to the

individual gas molecules in the plasma reactor.

2. Methods

2.1. Particle tracing

To obtain realistic trajectories of gas molecules flowing through a typical plasma reactor, as input to

our Monte Carlo calculations (section 2.2), we performed computational fluid dynamics (CFD)

followed by particle tracing simulations.

The CFD model describes the behaviour of a pure N2 gas flow in both an empty and a packed bed

cylindrical DBD reactor. Modelling pure N2, as opposed to a specific N2-H2 mixture, will deliver a

general overview for N2-containing gas flows, while avoiding the need for approximative theories for

calculating gas mixture properties like the viscosity [16]. The CFD model solves the mass continuity

and momentum continuity Navier-Stokes equations for an incompressible Newtonian fluid:

𝜌∇. 𝑢𝑔 = 0 (1)

𝜌(𝑢𝑔 ∙ ∇)𝑢𝑔 = ∇ ∙ [−𝑝𝐼 + (𝜇 + 𝜇𝑇) (∇𝑢𝑔 + ∇(𝑢𝑔 )𝑇)] + 𝐹 (2)

where 𝜌 stands for the gas density, 𝑢𝑔 is the gas flow velocity vector, superscript 𝑇 stands for

transposition, 𝑝 is the gas pressure, 𝜇 is the dynamic viscosity and 𝜇𝑇 the turbulent viscosity of the

fluid, 𝐼 is the unity tensor and 𝐹 is the body force vector.

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In an empty reactor, the flow is expected to be highly laminar, which greatly reduces the complexity of

equations 1 and 2, as we can consider 𝜇𝑇 = 0.

To be prepared for any turbulence occurring when a packing is added to the reactor geometry, the

turbulent properties of the flow were solved for in the packed bed DBD reactor. We achieved this by

using a Reynolds-averaged-Navier-Stokes (RANS) turbulent model, which significantly reduces the

computation time by averaging all fluctuating turbulent quantities (i.e., the turbulent kinetic energy and

the turbulent dissipation rate) over time. We used the k-ω model, as this model is still applicable at low

level of turbulence and is able to resolve the laminar boundary layer near the walls [17]. This is

important for the flow inside a packed bed reactor, as contact with the beads will introduce a lot of these

boundary layers.

To reduce the large computational time of the CFD calculations, especially regarding the complexity

due to the introduction of 13,216 beads inside the geometry, several measures were taken to reduce the

number of finite mesh elements in the modelled geometry. First, the axial symmetry allows us to

consider only 1/8th of the reactor, reducing the complexity of the model to 1,652 beads. Secondly, to

avoid the need for a very small finite element mesh to resolve the contact points between two beads,

the radius of the beads was reduced by 5 % for the flow calculation.

After the CFD calculations, we performed particle tracing simulations, in which we computed the

trajectory of gas molecules as they flow through the reactor. These trajectories are calculated based on

Newton’s law of motion, using the drag force imposed by the velocity fields that were previously

computed:

𝑑(𝑚𝑝𝑢𝑔 )

𝑑𝑡= 𝐹𝐷

(3)

where 𝑚𝑝 is the particle’s mass, 𝑢𝑔 is the gas flow velocity vector and 𝐹𝐷 the drag force.

We performed the trajectory calculations for 10,000 particles, i.e., gas molecules, to ensure statistically

relevant results. This yields 10,000 possible trajectories which the gas molecules can follow when

flowing through either the empty or the packed bed DBD reactor. However, the solver removes

trajectories from the simulation when the particles get stuck on a wall, due to their velocity which can

approximate 0 m/s upon collisions with the walls. This is especially quite significant in a packed bed

DBD where the particles often collide with the beads, but more than 5,000 particle trajectories were

still preserved in the simulation. Those remaining trajectories do also include collisions. The calculated

trajectories served as input to our Monte Carlo calculations (section 2.2) from which we determined the

fraction of microdischarges experienced by the gas molecules.

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Both the CFD and the particle tracing calculations were solved using the CFD module of COMSOL

version 5.5 [18]. We used the same residence time in the empty and packed bed reactor (see section

3.1.1).

The flow simulations do not account for plasma effects, because we consider a full scale packed bed

reactor with approximately 13,000 beads in the reactor (see above), and the complexity of a plasma

model on this scale would be too high.

2.2. Monte Carlo calculations

The Monte Carlo approach determines whether a particle along a specific trajectory through the reactor

(see section 2.1) is hit by a microdischarge, which occurs randomly throughout the reactor.

The total number of microdischarge events that take place in a DBD reactor is typically in the order of

a few millions. Because the microdischarges occur throughout the whole reactor and because single gas

molecules can only be at one point in the reactor at any moment in time, we expect that they experience

a reduced number of microdischarges. For this reason we employed a Monte Carlo approach to

statistically determine the fraction of microdischarges to which the individual gas molecules are

exposed. Hence, by the general Monte Carlo approach, we select random numbers between 0 and 1,

which determine at which location a microdischarge occurs. We assume that the locations of the

microdischarge events are uniformly distributed throughout the reactor gas volume (see next section for

motivation), but their actual occurrence is determined randomly. We performed experimental

diagnostics to show that a uniform distribution of the microdischarges throughout the reactor is indeed

a good approximation (see section 2.3 and 3.3). For more details on the placement of the microdischarge

events, we refer to section 3.2.

Furthermore, the microdischarge events are equally distributed over the gas residence time and the

microdischarges stay active for a certain microdischarge lifetime. For each trajectory through the reactor

we check whether or not a particle (i.e., gas molecule) is at the same location as a microdischarge at

any moment in time during which the specific microdischarge is active. If this is the case, we define a

hit for the microdischarge event. The total number of hits per trajectory over the total number of

microdischarges that took place in the reactor will then define the fraction of microdischarges.

We acknowledge that we do not capture the actual physics of a microdischarge, and we do not consider

an actual plasma, which could influence charged particles, especially in the microdischarges due to the

presence of strong electric fields. Indeed, self-consistent modelling of a filamentary plasma is a difficult

task, requiring e.g., particle-in-cell Monte Carlo methods [19]. As we consider millions of random

microdischarges instead of individual microdischarges, a fully self-consistent filamentary plasma model

is not feasible. However, we do apply this concept of the fraction of microdischarges in a 0D plasma

kinetics model (section 2.4).

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Finally, we note that our Monte Carlo calculation and the present study lead to general insights that are

not necessary specific to a single experimental setup. As long as the assumption on the uniform

distribution of the microdischarges is valid, the findings will be unaltered and our Monte Carlo

calculations should not have to be repeated. The conclusions are used in our 0D plasma kinetics model.

Thus overall, the modelling scheme and the modelling presented in this paper is quite different from

particle-in-cell Monte Carlo calculations. Indeed, the latter self-consistently describe the microscopic

behavior of individual particles, i.e., their collisions and movement in an electric field, which is

influenced by the space charge generated by the super-particles, hence completely different from our

model. Moreover, they typically focus on the detailed discharge behaviour of a single, or a few,

filaments, and not on the overall gas conversion process (like in our plasma kinetic modelling).

2.3. Experiments

In the Monte Carlo calculations we assume that the microdischarges are uniformly distributed

throughout the plasma reactor. We verified this assumption by means of experimental diagnostics.

The reactor used to study the distribution of microdischarges by ICCD images is a closed parallel plate

rectangular DBD reactor with two dielectrics. Each dielectric is a fused silica (quartz) piece of 118 x

68 x 3 mm3. A discharge gap of 2 mm is formed between them. On each outer side, the dielectrics are

covered by a stainless-steel mesh acting as electrode. The structural parts of the reactor, i.e., the

enclosure, are built in transparent polymethyl methacrylate (PMMA). This configuration allows to place

a high-speed camera perpendicular to the dielectric surface and to have a clear view of the discharge

(as illustrated in figure 1).

Figure 1. Schematic of the DBD setup used for ICCD imaging, showing the electrodes (A), dielectric barriers (B), discharge

gap (C), camera body (D), lens (E) and high voltage generator (F).

One of the electrodes is connected to a high voltage generator (AFS G10S-V) through a high frequency

transformer (1-30 kHz), while the other electrode is grounded. The generator allows to control the

operating frequency and the applied power in the plasma. The experiments were conducted at a

frequency of 12 kHz and a power of 100 W. Approximately 60% of this power is effectively absorbed

into the plasma. A constant flow rate of 1 L/min of a N2:O2 gas mixture with a ratio of 1:1 (500 mL/min

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each) is applied at atmospheric pressure. The high-speed camera used for ICCD imaging is a Photron

Nova S2 working at 10,000 frames per second (fps) at a raw resolution of 1024 x 672 pixels. The final

resolution is divided by two after application of a binning 2 x 2 to enhance the luminosity of the plasma,

giving a final resolution of 512 x 336 pixels.

Furthermore, we studied the effect of the fraction of microdischarges in a plasma kinetics model (see

section 2.4). We performed plasma-catalytic ammonia synthesis experiments to verify whether our

plasma kinetics model is realistic. The ammonia synthesis experiments were performed in an axial

cylindrical DBD reactor operated at atmospheric pressure, 23.5 kHz frequency and ca. 10 kV peak-to-

peak voltage, and a gas mixture of H2:N2 with a ratio of 1:1. The applied voltage correspond to 100 W

of supplied power of which approximately 70 % goes into the plasma. The catalyst was 10 wt%

Co/Al2O3 prepared via incipient wetness impregnation of commercial Al2O3 beads (diameter: 1.8 mm,

Sasol). The production of NH3 was monitored by mass spectrometry of the outlet gaseous mixture. The

full description of the reactor, plasma discharge, experimental procedure, analytical techniques, and

catalyst preparation and characterisation is found elsewhere [20].

In our experiments, the total gas flow rate was varied from 100 to 400 mL/min to allow different values

of the gas residence time inside the plasma region (see section 3.4).

2.4. Plasma kinetics model

We use our previous 0D plasma chemistry model [21] developed within ZDPlasKin [22] and catalytic

surface chemistry microkinetics model [21] to investigate the effect of using different fractions of

microdischarges and we compare the results against experiments (see section 2.3). In this model we

solved the continuity equation for the various plasma and surface species:

𝑑𝑛𝑖

𝑑𝑡= ∑𝑐𝑖,𝑟 (𝑘𝑟 ∏𝑛𝑙

𝑙

)

𝑟

(8)

where 𝑛𝑖 is the density of species 𝑖, 𝑐 is the stochiometric number of the species in reaction 𝑟, 𝑘𝑟 is the

rate coefficient and the subscript 𝑙 represents the species on the left hand side of the reaction. The rate

coefficients of the catalytic surface reactions are adopted from Engelmann et al. [20]. The gas phase

rate coefficients are adopted from literature and reported in [21]; they are constant or depend on the gas

temperature or electron temperature. The rate coefficients of electron impact reactions are evaluated

from electron impact cross sections using BOLSIG+ [23], which solves the Boltzmann equation for the

electrons to obtain the electron energy distribution function 𝑓(𝐫, 𝐯, 𝑡) (EEDF):

𝜕𝑓

𝜕𝑡+ 𝐯 ∙ ∇𝑓 −

𝑒

𝑚𝐄 ∙ ∇𝐯𝑓 = 𝐶 (9)

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where 𝑒 is the elementary charge, 𝑚 is the electron mass, 𝐯 is the velocity vector, ∇𝐯 is the velocity-

gradient and 𝐶 contains the actual electron impact collision terms; for more details, we refer to Hagelaar

and Pitchford [23]. The EEDF is calculated based on the collisions and electric field. The collisions are

defined by the cross sections, and in our model we provide the reduced electric field 𝐸/𝑁 based on the

power density 𝑃/𝑉 of the plasma [15], [21]:

𝐸 𝑁⁄ =1

𝑁√

𝑃/𝑉

𝑒𝑛𝑒𝜇𝑒

(10)

where 𝑁 is the total gas number density and 𝜇𝑒 is the electron mobility calculated from the EEDF [23].

Because of the calculation of the mobility, first an initial set value for the reduced electric field is used.

In order to capture the properties of a filamentary DBD, we describe the microdischarges as triangular

power density pulses with a certain lifetime, adopted from the experimental instantaneous plasma power

and an assumed microdischarge volume [21]. Such pulses indeed provide reduced electric fields,

electron densities and electron temperatures as expected for filamentary DBDs [8], [10]. We note that

in our model there is also an active plasma in between the microdischarges, but much weaker than the

actual microdischarges themselves. From the same data we determine the number of microdischarges

per half cycle (𝑁𝑀𝐷,½ 𝑐𝑦𝑐𝑙𝑒) [21]. Within the gas residence time in the reactor, we can then approximate

the total number of microdischarges in the reactor with:

𝑁𝑀𝐷,𝑡𝑜𝑡𝑎𝑙 = 2𝑁𝑀𝐷,½ 𝑐𝑦𝑐𝑙𝑒𝑓𝐷𝜏𝑟𝑒𝑠 (4)

where 𝑁𝑀𝐷 is the number of microdischarges in the whole reactor, during the full residence time

(subscript 𝑡𝑜𝑡𝑎𝑙) and during one discharge half cycle (subscript ½ 𝑐𝑦𝑐𝑙𝑒), 𝑓𝐷 is the discharge frequency

and 𝜏𝑟𝑒𝑠 is the residence time. This typically yields in the order of millions of microdischarges (i.e., by

using typical values of 𝑁𝑀𝐷,½ 𝑐𝑦𝑐𝑙𝑒 = 25, 𝑓𝐷 = 23.5 kHz and 𝜏𝑟𝑒𝑠 = 4 s [21]).

In plasma kinetics models of a DBD only a small fraction of those microdischarges are assumed to be

experienced by the molecules [8]–[15]. The reduced number of microdischarges that are actually

considered in the model can be described with:

𝑁𝑀𝐷,𝑟𝑒𝑑𝑢𝑐𝑒𝑑 = 𝜂𝑀𝐷𝑁𝑀𝐷,𝑡𝑜𝑡𝑎𝑙 (5)

where we introduce 𝜂𝑀𝐷, the fraction of microdischarges to which individual gas molecules are

exposed, which we determine in this study using the Monte Carlo approach (see section 2.2).

In our previous work we assumed that the fraction of microdischarges depends on the gas velocity and

the amount of reactor volume passed through by the gas molecules during one discharge period. This

caused the number of microdischarges in the model to be independent of the residence time [15]. In

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earlier work by our group PLASMANT, it was assumed that the fraction of microdischarges is given

by the microdischarge volume over the reactor volume [13], [14].

We will use this plasma kinetics model to investigate the effect of varying the fraction of

microdischarges over a large range (section 3.4).

3. Results and discussion

3.1. Particle tracing

3.1.1. Particle trajectories through an axial cylindrical packed bed reactor

We calculated particle trajectories through both an empty and packed bed cylindrically symmetrical

DBD reactor, as described in section 2.1. The flow rate in the empty DBD was adjusted relative to the

one in the packed bed DBD, to account for the reduced volume due to the packing in the latter, and to

obtain the same average residence time in both reactors. The empty reactor volume is approximately

9.42 cm3 and the packing volume is approximately 5.93 cm3 (i.e., 13,216 spheres with 1 mm diameter),

yielding a packing factor of 63 %. This results in a flow rate of 270 ml/min for the empty reactor, based

on a 100 ml/min flow rate in the packed bed reactor. Using those different flow rates, the average

residence time in both reactors is then 2.1 s. In figure 2 we show the packed bed reactor, the dimensions

of the packing, as well as the simulated gas molecules and their velocity. The figure shows that gas

molecules flow through the reactor at a wide range of velocities, varying from 0.002 m/s to 0.1 m/s.

Figure 2. Illustration of the gas molecules flowing through a packed bed reactor consisting of two concentric cylinders. The

reactor contains 13,216 spherical beads (1 mm in diameter) stacked into two layers in the gap between both concentric

cylinders. Note that the reactor diameter and reactor length are not to scale, for clarity. The inner diameter of the large cylinder

is 17 mm, while the outer diameter of the small cylinder is 13 mm, yielding a gaseous gap of 2 mm.

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3.1.2. Residence time distribution in an empty and packed bed DBD reactor

In figure 3a and 3b, we show the residence time distribution of the gas based on the particle tracing

calculations for an empty and a packed bed DBD reactor, respectively. Besides the tail of the

distribution, which is less smooth in case of the packed reactor, the distributions are quite similar.

Indeed, the flow rate was chosen to yield the same average residence time in both reactors (see section

3.1.1).

The gas flow in an empty reactor follows a typical Poiseuille velocity profile, as illustrated in figure 4a.

As a result of the fluid viscosity, friction between the flowing gas and the reactor wall causes the gas to

slow down close to the wall, creating a boundary layer that shapes the parabolic velocity profile. Hence,

the molecules in vicinity to the reactor walls in the empty reactor are characterized by a longer residence

time compared to the average. The molecules in the middle of the reactor have a high velocity and

account for the shorter residence times, resulting in the distribution as shown in figure 3a.

The gas flow in the packed reactor shows opposite behaviour, as illustrated in figure 4b. Fast molecules,

with short residence times, now flow through the reactor in the vicinity of the wall, while the slow

molecules, with long residence times, flow through the middle of the reactor. This effect is discussed

in greater detail below (section 3.1.3).

Figure 3. Residence time distribution in the empty (a) and packed bed (b) reactor. The average residence time is the same in

both reactors, and is indicated with a vertical dashed line (approximately 2.1 s).

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Figure 4. Top view of the axial velocity of each particle in the empty (a) and packed bed (b) reactor. The white space outside

of the packing bead contours in (b) represents locations where the probability of finding a particle is so low that none of the

simulated particles were found there. Note, both figures show a cross section of the reactor made at a specific axial height (z

= 90 mm, i.e., 10 mm before the reactor outlet). In (b) the intersection of this plane with the packing beads does not cut the

beads equally because the two layers are not aligned in a closed packing of equal spheres, hence the outer layer of beads

exhibits a smaller radius.

3.1.3. Gas distribution in an empty and packed bed DBD reactor

When a packed bed is introduced in the reactor, the incoming gas is forced to flow through the small

gaps in between the beads of the packed bed, which drastically changes the distribution of the gas flow

compared to an empty reactor. In figure 5 we present the gas distribution for an empty and a packed

bed DBD reactor, as calculated by the particle tracing simulations. For an empty reactor, figure 5a

shows that most of the gas passes through the centre of the reactor, where the flow is undisturbed by

the viscous boundary layer near the walls. As shown by figure 5b, the introduction of a packed bed

seemingly inverts this profile, as the flow is redistributed towards the reactor walls. The gas flow

favours the path of least resistance and thus avoids the darkest blue zones around r = 7.1 mm and r =

7.9 mm, where the two bead layers are located (cf. figure 4b). The small gaps between the beads only

allow for small flow rates through the bed, so approximately 90% of the gas molecules flows towards

the outer layer of the bed and the reactor wall, where the gaps are larger than in the perfectly packed

centre. This value is based on integrating the probability distribution at the end of the reactor (figure 5b

at z = 100 mm) and comparing the outer layers to the middle section (between a radius of 6.9 and 7.9

mm), where the average probability of finding a gas molecule is 0.006 compared to 0.1 in the outer

layers.

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Figure 5. Radial gas distribution as a function of axial position in the reactor (i.e., along the reactor length) in the empty (a)

and packed bed (b) reactor. The radial distribution was normalized at each axial position in the reactor, so integration along a

vertical line in the figure equals to 1.

Figure 5a and 5b both show that the gas density near the outer wall (towards 8.5 mm radius) is slightly

higher than the gas density near the inner wall (6.5 mm radius). This is due to the curve of the cylindrical

DBD reactor. Indeed, there can be more gas molecules at the larger radius due to the larger volume (or

circumference). For the packed bed reactor this effect is larger, as also the packing configuration has to

be considered. This is illustrated in figure 6, which shows the close packing of equal spheres for a

packed bed DBD reactor both with a straight and a curved electrode. The figure demonstrates that a

curved electrode introduces more space (highlighted in red) between the beads in the upper bead layer,

allowing for more gas to flow through.

Figure 6. Close packing of equal spheres for a packed bed DBD reactor with a straight (a) and a curved (b) electrode surface.

The fact that the incoming gas is not evenly distributed over the reactor bed could be an important

consideration for catalytic processes in packed bed DBD plasma reactors. Only a smaller portion of the

gas will flow through the middle of the bed, i.e., in between the two bead layers, where there is more

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contact with the catalyst bed, and a longer residence time in the plasma (see previous section). However,

in reality, the ideal packing assumed in this model is unlikely due to the bead shape and size and the

filling of the reactor. Thus, in reality, gaps of various sizes can be formed throughout the packed bed,

which can also allow more gas to flow through the centre of the bed.

3.2. Monte Carlo calculations

3.2.1. Microdischarge distribution and “hits” in the empty and packed bed reactor

We calculated the fraction of microdischarges with the Monte Carlo approach explained in section 2.2,

which we applied to the particle trajectories of the empty and packed bed DBD reactor (cf. section 3.1).

We assume 25 microdischarges per discharge half cycle, with a lifetime of 200 ns and a discharge

frequency of 23.5 kHz [21]. With the maximum residence time of 9.41 s (corresponding to the empty

reactor, cf. figure 4a), this gives 10,739,500 microdischarges taking place throughout the whole reactor,

during the entire gas residence time in the reactor. The maximum residence time in the packed reactor

was 8.63 s, corresponding to 10,140,250 microdischarges.

In the empty reactor, we assumed the microdischarges to be small channels between the inner and outer

electrode. For simplicity, the channels are actually very thin circle sector, which allows us to use the

cylindrical coordinate system throughout the Monte Carlo calculations. The azimuthal width of the

microdischarges is 0.1 rad and the axial width is 0.1 mm. The channel length is approximately 2 mm,

corresponding to an approximate microdischarge volume of 0.15 mm3. For clarity, the definition of the

circle sector and the thin microdischarge channel is shown in figure 7. The microdischarge events are

placed randomly (as defined by the Monte Carlo approach) throughout the full reactor gas volume.

Figure 7. Schematic showing a top view of approximately 1/4th of the cylindrical reactor (dashed black lines), the geometrical

definition of a circle sector (red outline), the modelled reactor segment (black outline) and a thin circle sector representing a

microdischarge channel in the empty reactor (blue outline).

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In the packed bed reactor, the microdischarge events are defined as spheres with 0.3 mm radius,

corresponding to an approximate microdischarge volume of 0.11 mm3 [21]. The events are uniformly

distributed throughout the packed bed reactor by randomly choosing a packing bead and then randomly

choosing a point on the surface of this specific bead. Filamentary DBDs, especially with packed beds,

are often characterized by point-to-point discharges, i.e., microdischarges between two beads, or surface

discharges, i.e., microdischarges over the surface of a packing bead [24]. Our chosen spherical

microdischarge volumes, which we centre on the surface of the packing beads, can be considered a

combination of both, as the chosen radius is also large enough to bridge the gaps between two close

packing beads.

In figure 8, we illustrate the locations of all the microdischarge events throughout the reactors. Note

that this figure only shows the locations and not the size (0.1 mm width and 0.3 mm radius in case of

the empty and packed bed reactor, respectively) for the sake of clarity.

Figure 8. The microdischarge locations in both the empty (a) and the packed bed (b) DBD reactor. Only 1/10th of the reactor

length is shown, with all of the microdischarges taking place (dark areas created by the overlapping and connecting individual

vertical lines (a) or dots (b)). The red lines and dots indicate microdischarge events that have hit a particle trajectory. Note that

in (a) only 1 % of all microdischarges are plotted for the sake of clarity. In addition, because in (b) the microdischarge events

are placed on the surface of the packing beads, the shape of the packing beads is reflected back in this figure.

In figure 9, we show five randomly selected particle trajectories and only the microdischarges to which

they were exposed in the empty and packed reactor. It is clear that a particle trajectory can experience

a microdischarge event anywhere along its trajectory. In case of the packed bed reactor (figure 9b and

9c) particles (i.e., gas molecules) travelling near the walls are typically exposed to only a few

microdischarges along their trajectory, while particles that travel through the centre of the packed bed

reactor experience more microdischarges. This is due to the longer residence time, as discussed in

14

section 3.1, and because there are somewhat more microdischarge events in the middle of the reactor

(at a radius of 7.5 mm), where the two packing bead layers are in contact with each other (cf. figure 8b,

which showed all microdischarges that took place, also including microdischarge events that did not hit

any of the thousands of traced particles).

Figure 9. Five randomly selected particle trajectories flowing through the empty reactor (a) and the packed bed reactor (b, c),

as well as the microdischarges to which they were exposed (vertical lines or spheres in the same colour). In (a) and (b) the full

reactor length is shown (reactor length and radius not to scale). In (c) 1/10th of the reactor length is shown (with reactor length

and radius to scale), i.e., between z = 10 and 20 mm, where most of the hits occur for the orange and yellow trajectory, because

those particle trajectories go here through the middle of the packed bed, where they move more slowly (cf. figure 4b), and

there are slightly more microdischarge events.

In figure 10, we show that the hits between gas molecules and microdischarge events are also randomly

distributed in time. We show this for the case of the packed bed, but it is also true for the empty reactor.

Any gas molecule can have relatively short and long times between two microdischarges (cf. figure 10

for the same trajectories as in figure 9b). Note that the orange and yellow particle trajectories, which

pass through the middle of the packed bed between z = 10 and 20 mm (cf. figure 9b) have a longer

residence time (3.12 s and 3.03 s) and experience more microdischarges.

15

In section 3.1.2 and 3.1.3 we explained that most of the particles travel through the packed bed reactor

in the outer layers (cf. figure 4b and 5b). Thus, only a few particles travel through the centre of the

reactor, and experience a large number of microdischarges (cf. orange and yellow particle trajectories

in figure 9b and 9c). Most particles will travel through the reactor near the electrode walls and

experience only a few microdischarges.

Figure 10. The moments in time at which the microdischarge event hits of figure 9b occurred (dots on the line) for each of the

five particle trajectories. The legend indicates the residence time of each displayed trajectory.

The distribution of the time between two microdischarge hits, based on all trajectories and all successful

microdischarge events, is given in figure 11. The median of this time distribution is 47 ms, the mode is

67 ms and the mean is 74 ms, the time between two microdischarge hits ranges from 68 μs to 1.1 s.

Figure 11. Distribution of the time between two microdischarge hits, when considering all trajectories. The vertical lines

indicate, from left to right, the median (47 ms), mode (67 ms) and mean (74 ms), respectively.

3.2.2. Determination of the fraction of microdischarges to which the gas molecules are exposed

We use the results of each trajectory to determine the fraction of microdischarges to which the gas

molecules are exposed, by fitting the Monte Carlo results of both the empty and packed bed reactor (see

figure 12) to:

𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑖𝑡𝑠 = 𝑥1 × 𝜏𝑟𝑒𝑠 (6)

where 𝑥1 is the slope of the fit, from which the fraction of microdischarges can be evaluated based on

equation 4 and 5 above:

𝜂𝑀𝐷 =𝑥1

2𝑁𝑀𝐷,½ 𝑐𝑦𝑐𝑙𝑒𝑓𝐷(7)

16

The fits are shown in figure 12. For each of the particle trajectories, we plot the number of

microdischarge hits and their residence time. Each cross in figure 12 represents a particle trajectory.

The slope of the fit through this plot of number of microdischarge hits vs the residence time gives us

the value 𝑥1 from equation 6 and from this slope we determine the fraction of microdischarges (equation

7).

Based on the slopes in figure 12 (i.e., the values of 𝑥1) and equation 7, we calculate the fraction of

microdischarges to which the gas molecules are exposed, as (1.548 ± 0.003)×10-5 and (1.306 ±

0.006)×10-5, for the empty and packed bed reactor, respectively.

Note that the linear fits go through the origin of the graphs, based on the expected relationship (cf.

equation 6), but at longer residence times the fit does not seem optimal when enforcing this intersection

at (0,0) for the packed bed reactor (figure 12b). Indeed, the points at longer residence time tend to lie

above the fitted line. This is attributed to the fact that the slower particles move through the middle of

the packed bed (cf. section 3.1), where there are slightly more microdischarge events, because the two

packing bead layers are in contact with each other, hence explaining the slightly higher number of hits

compared to the fitted line.

Figure 12. Number of microdischarge hits as a function of average residence time, for each of the particle trajectories, for the

empty reactor (a) and the packed bed reactor (b). Each point corresponds to one trajectory. The data is fitted to equation 6, and

the slope of the fit (𝑥1) is also indicated.

17

3.2.3. Empirical relationship for the fraction of microdischarges

As mentioned before, in earlier work from our group we assumed that the fraction of microdischarges

to which the gas molecules are exposed, 𝜂𝑀𝐷, is given by the microdischarge volume over the reactor

volume [13], [14]. Based on the reactor volume 𝑉𝑟𝑒𝑎𝑐𝑡𝑜𝑟, the packing factor 𝛼 and the microdischarge

volume 𝑉𝑀𝐷 used in the COMSOL model and Monte Carlo calculations, we can also try to evaluate the

fraction of microdischarges based on such volume-based relations:

𝜂𝑀𝐷,𝑒𝑚𝑝𝑡𝑦 =𝑉𝑀𝐷

𝑉𝑟𝑒𝑎𝑐𝑡𝑜𝑟

(11)

𝜂𝑀𝐷,𝑝𝑎𝑐𝑘𝑒𝑑 =𝑉𝑀𝐷

(1 − 𝛼)𝑉𝑟𝑒𝑎𝑐𝑡𝑜𝑟

(12)

With the microdischarge volume of 0.15 mm3 and 0.11 mm3 for the empty and packed bed reactor,

respectively, the reactor volume of 9.42 cm3 and the packing factor of 63 %, equation 11 and 12

calculate the fraction of microdischarges as 1.6×10-5 and 3.2×10-5 for the empty and packed bed reactor,

respectively.

Hence, using this method, the fraction of microdischarges for the empty reactor is in very good

agreement with the Monte Carlo results (i.e., 1.6×10-5 vs. (1.548 ± 0.003)×10-5. On the other hand, the

volume-based fraction of microdischarges for the packed bed reactor is somewhat overestimated

compared to the Monte Carlo calculation (i.e., 3.2×10-5 vs (1.306 ± 0.006)×10-5). However, in this case

we need to take into account that the effective microdischarge volume used in the Monte Carlo

calculations is somewhat lower than 0.11 mm³, due to the placement of the spherical microdischarge

events on the surface of the packing beads and because the assumed microdischarge radius is larger

than the smallest distance between the packing beads. If we account for this in equation 12 (by assuming

𝑉𝑀𝐷/2 instead of 𝑉𝑀𝐷), we obtain a fraction of microdischarges of 1.6×10-5, which shows a much better

agreement with the Monte Carlo results. We note that the fraction ½ is just an approximation. Indeed,

in our Monte Carlo calculations for the packed bed reactor, we did place the spherical microdischarge

events with an actual size of 0.11 mm3 and the exact effective microdischarge volume is difficult to

determine due to the complexity of the geometry of the packing beads and the stochastic nature of the

calculation. The empty reactor (discussed above) is a much simpler case, as only the reactor walls give

rise to a smaller effective microdischarge size.

It is clear that, in the case of a packed bed reactor, a direct comparison between the proposed simple

relationship (of microdischarge volume over reactor volume) and the Monte Carlo calculations applied

to a realistic reactor set-up is difficult, due to the definition of the microdischarge events, both in terms

of their size and where they take place. However, the fact that the simple relationship predicts values

close to the more explicit derivation based on the particle trajectories and Monte Carlo calculations,

18

especially for the empty reactor, is quite striking. Hence, we believe that the fraction of microdischarges

to which the gas molecules are exposed, can be properly approximated by the microdischarge volume

over the reactor gas volume.

Such a relationship between the microdischarge volume, reactor gas volume and fraction of

microdischarges experienced by the molecules can be understood by the following explanation. If the

microdischarges can occur throughout the reactor with the same probability, we can consider that, on

average, the microdischarges always take place in the same point of the reactor. This is true for any

point in the reactor and thus also any point that will be crossed by a gas molecule.

Multiple microdischarges occurring at the same time would not increase the probability for gas

molecules of being hit by a microdischarge. Indeed, when multiple microdischarges occur at the same

time but at different places, it is clear that a molecule can only experience one of these microdischarges

at most.

3.3. Experimentally observed microdischarge distributions

In the previous section, we calculated the fraction of microdischarges based on a Monte Carlo approach,

assuming a uniform distribution of the microdischarges. Various authors have recorded DBD plasma in

both empty reactors [3] and packed bed reactors [24]–[27]. Wang et al. specifically reported their DBD

plasma to become more uniform upon introducing a packing [28].

We made ICCD recording of a filamentary DBD plasma in an empty reactor. A transparent top electrode

and dielectric allowed to record the microdischarges from above, as depicted in figure 13. We show a

standard photograph of the plasma (a), as well as the ICCD recording over a single frame (b) and

multiple frames (c,d). We note that the microdischarge lifetime (in the order of 10 ns) is much smaller

than the time window of a single frame (0.1 ms).

Figure 13. Photograph of a filamentary DBD plasma, driven at 12 kHz and 100 W (a), and ICCD recording of a single frame

(0.1 ms, b) and multiple frames (i.e., 1,000 frames, 0.1 s (c) and 3,000 frames, 0.3 s (d)). The size of the grid is 10 x 10 mm2.

19

From figure 13 we can conclude that a uniform distribution of microdischarges is indeed an adequate

assumption when considering typical time scales associated with the residence time (i.e., seconds). We

note however that the exact behaviour may depend on the discharge frequency, which will affect the

randomness of the microdischarges due to the memory effect, leading to the self-organization of

microdischarges as reported in literature [29], [30]. Those effects were also observed in our own

experiments (not shown), but we note that such self-organized microdischarge distributions typically

still cover the whole reactor volume, which means that this effect does not significantly impact the

assumption behind our Monte Carlo calculations. Also the gas flow can influence the microdischarge

location [30]. This can also be seen in figures 13c and 13d, where microdischarges follow a path in the

direction of the gas flow (from left to right) creating aligned dots instead of random dots (indicated in

the red boxes). This effect was observed to be more pronounced at higher discharge frequencies, which

is also attributed to the memory effect. The microdischarges moving with the gas flow, might be an

effect that increases the total duration during which a gas molecule is exposed to the strong

microdischarges. In general, we judge that our assumptions on the uniform distribution of

microdischarges, as used in the Monte Carlo calculations, is valid from 4 to 30 kHz, based on our

experimental observations.

Finally, the operating pressure could influence the above observations and thus the assumptions behind

the Monte Carlo calculations should be re-evaluated for such cases, i.e., low pressure conditions.

However, low pressure plasmas are typically more homogeneous and thus do not exhibit the filamentary

microdischarges of interest [31]. The memory effect is typically also a function of pressure due to the

change in mean free path of the gas particles [32].

3.4. Experimental and calculated plasma-catalytic NH3 yield

We now want to verify whether the concept of the fraction of microdischarges, as calculated in section

3.2, really has a physical meaning and whether our proposed relationship (i.e., the fraction of

microdischarges is the ratio of microdischarge volume over the reactor gas volume) provides a realistic

value. Therefore, we applied this concept to plasma-catalytic NH3 synthesis, using our previously

developed plasma chemistry and catalytic surface chemistry microkinetics models [21], by running

them as a function of residence time, and comparing with plasma-catalytic NH3 synthesis experiments

in a packed bed DBD with Co catalyst on Al2O3 beads, at different residence times.

We now adopt the exact microdischarge volume of 0.088 mm3 of our previous study, which is also

related to the plasma power density in the model [21]. The relevant reactor volume after packing was

6.4 cm3, which gives a fraction of microdischarges of 1.37×10-5. Again assuming 25 microdischarges

per discharge half cycle and a discharge frequency of 23.5 kHz, following our experimental conditions,

gives a microdischarge frequency of 1,175,000 s-1 (i.e., 2𝑁𝑀𝐷,½ 𝑐𝑦𝑐𝑙𝑒𝑓𝐷, cf. also equation 4) and 16 s-1

after applying the above fraction of microdischarges (i.e., 𝜂𝑀𝐷2𝑁𝑀𝐷,½ 𝑐𝑦𝑐𝑙𝑒𝑓𝐷, cf. also equation 5). The

20

simulation provides the NH3 concentration as a function of time, for a total residence time of 3.84 s

[21], which corresponds to 61 microdischarges that will be considered in the model, based on the above

fraction of microdischarges. The experiments were performed for different flow rates, ranging between

100 and 400 ml/min, thus yielding different residence times ranging from 1.0 to 3.8 s, which can directly

be compared to the time dependence in the model.

In figure 14, we compare the experimental results against the calculated results, obtained by performing

a sensitivity analysis by varying the fraction of microdischarges over four orders of magnitude (between

10-6 and 10-2, figure 14a) and within the same order of magnitude (between 0.5×10-5 and 9.0×10-5, figure

14b).

Figure 14. Calculated NH3 concentration as a function of residence time, for different fractions of microdischarges, compared

to experimentally obtained values (Exp.). The fraction of microdischarges is varied over four orders of magnitude in (a) and

within the same order of magnitude in (b).

It is clear that a much larger or smaller fraction of microdischarges gives unrealistically fast and slow

NH3 synthesis rates (cf. figure 14a). This confirms that the order of magnitude of the fraction of

microdischarges, as obtained by our combined particle tracing and Monte Carlo simulations, appears

realistic. Figure 14b shows that the results are quite sensitive to the assumption of the fraction of

microdischarges, i.e., even a factor 2 makes a large difference. The experimental data agrees best with

a fraction of microdischarges of 0.9×10-5, which is slightly lower than the value of 1.37×10-5, based on

the assumed microdischarge volumes [21]. However, we note that the chosen discharge volumes, the

21

resulting power density and thus the plasma kinetics are subject to uncertainties. Nevertheless, the

agreement is still very reasonable; the fraction of microdischarges yielding best agreement with the

experiments is close to the value predicted by the empirical relationship (microdischarge volume over

reactor gas volume) as discussed in section 3.2, and the value resulting from the combined particle

tracing and Monte Carlo modelling approach. This indicates that our combined particle tracing and

Monte Carlo simulations method can provide a realistic picture of the fraction of microdischarges

experienced by the gas molecules.

In figure 15 we plot the NH3 concentration at the end of the reactor as a function of the fraction of

microdischarges, with the latter both on a linear scale and logarithmic scale (inset, to cover a larger

range). We can see that the NH3 concentration increases with the fraction of microdischarges, but for a

larger fraction of microdischarges (between 10-4 and 10-3), it remains fairly constant, and for still larger

fractions, it starts to drop (see inset in the figure). This is attributed to the interpulse time. When

molecules experience more microdischarges (corresponding to a large fraction of microdischarges), the

interpulse time is smaller (e.g., in the order of 100 ms and 100 μs for a fraction of microdischarges of

10-5 and 10-2, respectively). In our previous study, we found that NH3 is net produced in between the

microdischarges and net destroyed during the microdischarges themselves. However, one so-called

microdischarge and afterglow pair causes a net NH3 gain, until a steady state is reached [21]. Based on

these considerations, we indeed expect that there is an optimal interpulse time, which we find between

1 and 10 ms (i.e., a microdischarge fraction between approximately 10-3 and 10-4, respectively) based

on figure 15.

Figure 15. Calculated NH3 concentration at the end of the reactor as a function of the fraction of microdischarges on both a

linear scale and logarithmic scale (inset, with 2 extra data points (red dots)). A microdischarge fraction of 1.37×10-5 is indicated

with a 1 % and 10 % spread (dark and light red areas, respectively).

The Monte Carlo calculations resulted in fractions of microdischarges with typical uncertainties, less

than 1 %. However, we note that this is the average behaviour and that the gas could also be described

with a microdischarge fraction distribution, by evaluating the fraction of microdischarges for each

22

particle trajectory individually. Assuming the average fraction of microdischarges of 1.37×10-5 and an

uncertainty of 1 % gives an NH3 concentration of 9880 ± 90 ppm (cf. figure 15). A difference of 3, 5 or

10 %, which could be relevant when considering an actual microdischarge fraction distribution, would

give a difference of 260, 430 and 870 ppm, respectively.

3.5. Generalization of the concept: fraction of plasma power transferred to the gas molecules

The fraction of microdischarges experienced by the molecules can be considered as a plasma

characteristic of a filamentary DBD. At the same time, however, this quantity directly represents the

fact that not all power put into the plasma is equally transferred to each molecule [15]. In other words,

gas molecules only see a fraction of the power put into the plasma. This is a direct consequence of the

non-uniformity and stochastic behaviour of the plasma. Gas molecules would only be able to experience

all the plasma power if a plasma is perfectly uniform (i.e., constant plasma power density) in space and

time, and if it occupies the entire reactor, such that all gas molecules passing through the reactor would

effectively pass through the plasma region and experience the same plasma power. Either plasma non-

uniformity or stochastic behaviour of e.g., microdischarges will reduce the fraction of power transferred

to the individual molecules, as these effects give rise to individual gas molecules being treated

differently by the plasma.

As discussed above, the fraction of microdischarges can be generalized to the fraction of power

transferred to each molecule, and this quantity can in principle be defined for any plasma and is related

to both spatial and temporal plasma (non)uniformity. It can be considered as a kind of “plasma reactor

efficiency”. For instance, in a gliding arc (GA) plasma, only a limited fraction of the gas molecules

actually passes through the active arc plasma, as the latter does not occupy the entire reactor volume,

and this limits the “plasma reactor efficiency” [33]–[35]. The same was observed in an atmospheric

pressure glow discharge (APGD) [36]. For a DBD, this fraction of power transferred would correspond

to the fraction of microdischarges experienced by the molecules, giving rise to a very low “plasma

reactor efficiency” (i.e., order of 0.001 %). However, this reasoning assumes that the entire applied

plasma power is deposited in the microdischarges. In our recent study on plasma-catalytic NH3

synthesis, we considered typical current-voltage characteristics of our packed bed DBD and found that

a significant portion of the applied power still gives rise to a uniform plasma power (related to surface

discharges at the packing beads [24]). In other words, only part of the input power is transferred to the

microdischarges. In that case, the “plasma reactor efficiency” can be even close to 50 % [21]. This could

explain the higher efficiency of a packed bed DBD (characterized by a combination of filamentary and

surface microdischarges) compared to an empty DBD (characterized by only filamentary

microdischarges), as often reported in plasma-catalytic experiments [37]–[41].

Finally, we note that a filamentary plasma, which, according to the above discussion, would have a

lower “plasma reactor efficiency”, typically exhibits much stronger plasma conditions compared to

23

uniform plasmas, which would correspond to higher “plasma reactor efficiencies”. This means that both

types of plasmas are of interest in their own way, and warrant further investigation, due to completely

different underlying mechanisms and chemical kinetics.

4. Conclusions

We performed particle tracing simulations through both an empty and packed bed DBD reactor,

typically used for plasma catalysis. In the packed bed reactor, we placed approximately 13,000 hard

sphere beads in the gap between two concentric cylinders. We discussed in detail the flow

characteristics and residence time distribution of the gas molecules in both an empty and packed bed

DBD. We showed that the addition of a packed bed creates viscous boundary layers that slow down the

gas in the centre of the packed bed where two packing bead layers are near to each other. Furthermore

we showed that nearly 90% of the gas is redistributed towards the wall of the reactor, where there is a

relatively large distance between the reactor wall and the packing beads.

The calculated particle trajectories were used in a Monte Carlo calculation, mimicking a filamentary

DBD, to determine the fraction of microdischarges experienced by a single gas molecule. The results

provided evidence for an empirical expression of this fraction of microdischarges, that is, the fraction

of microdischarges is roughly equal to the microdischarge volume over the reactor gas volume.

To verify whether this proposed relationship (i.e., the fraction of microdischarges is the ratio of

microdischarge volume over the reactor gas volume) provides a realistic value, we performed plasma

and catalyst surface kinetics simulations for plasma-catalytic NH3 synthesis, in which the fraction of

microdischarges was used as input, and we compared the calculated NH3 concentration as a function of

time with experimental data. The experimental results could only be reproduced when the fraction of

microdischarges was consistent with the definition based of microdischarge volume over reactor gas

volume. The modelling results were highly sensitive to the assumed fraction of microdischarges, but a

good agreement was obtained with the predicted value from the Monte Carlo simulations, which

validates our concept of the fraction of microdischarges.

Finally, we can generalize the fraction of microdischarges, as it actually represents the fraction of

plasma power transferred to the gas molecules. Indeed, not all power will be transferred to each

molecule when the plasma is spatially and temporally non-uniform, and this principle is valid for other

plasma types than DBDs as well, such as GA plasma or APGD. Based on these principles, we can define

a “plasma reactor efficiency”, which in case of a DBD plasma can be directly related to the fraction of

microdischarges.

Acknowledgements

This research was supported by the Excellence of Science FWO-FNRS project (FWO grant ID

GoF9618n, EOS ID 30505023), the European Research Council (ERC) under the European Union’s

24

Horizon 2020 research and innovation programme (grant agreement No 810182 – SCOPE ERC

Synergy project) and by the Flemish Government through the Moonshot cSBO project P2C

(HBC.2019.0108). The calculations were performed using the Turing HPC infrastructure at the CalcUA

core facility of the Universiteit Antwerpen (UAntwerpen), a division of the Flemish Supercomputer

Center VSC, funded by the Hercules Foundation, the Flemish Government (department EWI) and the

UAntwerpen. The authors would also like to thank Hamid Ahmadi Eshtehardi for discussions on the

plasma-kinetic DBD model and Yannick Engelmann for discussions on the surface kinetics model.

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